本经验介绍含2π-α诱导类型三角函数的不定积分，即求∫sin(2π-α)dα，∫cos(2π-α)dα，∫tan(2π-α)dα，∫cot(2π-α)dα，∫sec(2π-α)dα，∫csc(2π-α)dα的步骤。
工具/原料
三角函数基本知识
不定积分基本知识
1.含2π-α的诱导公式
1、sin(2π-α)=-sin αcos(2π-α)=cos αtan(2π-α)=-tan αcot(2π-α)=-cot αsec(2π-α)=sec αcsc(2π-α)=-csc α
2、图例解析如下：
2.sin(2π-α)的不定积分
1、∫sin(2π-α)dα=-∫sin(2π-α)d(2π-α)=cos(2π-α)+c=cosα+c
2、图例解析如下：
3.cos(2π-α)的不定积分
1、∫cos(2π-α)dα=-∫cos(2π-α)d(2π-α)=-sin(2π-α)+c=sinα+c
2、图例解析如下：
4.tan(2π-α)的不定积分
1、∫tan(2π-α)dα屏顿幂垂=-∫tan(2π-α)d(2π-α)=-∫[sin(2π-α)d(2π-α)/ cos(2π-α)柯计瓤绘]=∫d cos(2π-α)/cos(2π-α)=ln|cos(2π-α)|+c=ln|cosα|+c
2、图例解析如下：
5.cot(2π-α)的不定积分
1、∫cot(2π-α)dα屏顿幂垂=-∫cot(2π-α)d(2π-α)=-∫[cos(2π-α)d(2π-α)/ sin(2π-α)柯计瓤绘]=-∫d sin(2π-α)/sin(2π-α)=-ln|sin(2π-α)|+c=-ln|sinα|+c
2、图例解析如下：
6.sec(2π-α)的不定积分
1、∫sec(2π-α)dα屏顿幂垂=-∫d(2π-α)/ cos(2π-α)=-∫cos(2π-α)d(2π-α)/ [cos(2π-珍提疮翘α)]^2=-∫dsin(2π-α)/ [1-(sin(2π-α))^2}=-∫dsin(2π-α)/ [(1-sin(2π-α))(1+ sin(2π-α))]=-(1/2)[∫dsin(2π-α)/ (1-sin(2π-α))+∫dsin(2π-α)/ (1+sin(2π-α))]=-(1/2)ln{[1+sin(2π-α)]/ [1-sin(2π-α)]}+c=-(1/2)ln[(1+sin(2π-α))/(1-sin(2π-α))]+c=-(1/2)ln[(1+sin(2π-α))^2/(cos(2π-α))^2]+c=-ln|(1+sin(2π-α))/cos(2π-α)|+c=-ln|(1-sinα)/cosα|+c=-ln|secα-tanα|+c
2、图例解析如下：
7.csc(2π-α)的不定积分
1、∫csc(2π-α)dα屏顿幂垂=-∫csc(2π-α)d(2π-α)=-∫d(2π-α)/ sin(2π-α)=-∫sin(2π-珍提疮翘α)d(2π-α)/ [sin(2π-α)]^2=∫dcos(2π-α)/ [1-(cos(2π-α))^2]=∫dcos(2π-α)/ [(1-cos(2π-α))(1+ cos(2π-α))]=(1/2)[∫dcos(2π-α)/ (1-cos(2π-α))+∫dcos(2π-α)/ (1+cos(2π-α))]=(1/2)ln[(1+cos(2π-α))/ (1-cos(2π-α))]+c=(1/2)ln[(1+cos(2π-α))^2/(sin(2π-α))^2]+c=ln|(1+cos(2π-α))/sin(2π-α)|+c=ln|(1+cosα)/sinα|+c=ln|cscα+cota|+c
2、图例解析如下：
